Synchronous oscillations in the central nervous system represent coordinated neural activity in networks that involve extensive feedback interactions whose role is difficult to analyze. In contrast, it is experimentally feasible to measure the dynamic responses of neurons and synapses to obtain feed-forward relationships that describe how specific output variables, such as synaptic strength or dynamics, depend on input parameters such as network frequency. We propose to develop a theoretical framework in which such feed-forward relationships, as measured in a central pattern generator (CPG) oscillatory network, are combined to build recursive feedback maps-functions that describe how the variables of the oscillatory network in one cycle can be determined from those in the previous cycles. The crustacean pyloric networkis a well-studied CPG with known synaptic connectivity and provides an excellent system to develop such a framework. To characterize the dynamics of the pyloric neurons and synapses we focus on the following three categories of experiments: 1. We have previously explored the role of short-term synaptic dynamics in shaping the pyloric network output and found that pyloric pacemaker neurons and synapses have preferred (resonance) frequencies at which they respond maximally. To explore the role of synaptic and membrane resonance, we will examine the following hypotheses: A. The membrane resonance frequency of pyloric neurons can bias the network frequency through gap-junction or chemical synaptic coupling. B. Synaptic resonance increases network stability. C. Membrane and synaptic resonance are subject to neuromodulation which can change their roles in network function. D. Modulation of membrane resonance of pyloric pacemaker neurons by proctolin reduces variability of network frequency. 2. We will measure how the peak phases of pyloric synapses and the burst onset and termination phases of synaptically-isolated neurons depend on input frequency and the neuronal oscillation waveform shape and amplitude. 3. We will measure the phase response curves of the pacemaker neurons in response to synaptic inputs with distinct peak phases in control conditions and in the presence of the modulatory peptide proctolin. The hypothesis that the response properties of a bursting neuron changes when the neuron is synaptically embedded in a network will be examined. The feed-forward relationships characterized in these experiments will be described using biophysical computational models and their function in network activity will be examined using the dynamic clamp technique. From these feed- forward relationships, we will build progressively more accurate feedback maps which will be mathematically analyzed. Stable equilibrium points of such maps correspond to stable rhythmic activity and can therefore be used to determine parameters that are important for the existence and stability of the pyloric network oscillations. Because factors that change the properties of neurons and synapses, such as extrinsic neuromodulation, alter the feed-forward relationships, their effect on network oscillations can be determined by analyzing the changes in the recursive feedback maps.